On the One Dimensional Poisson Random Geometric Graph

2008

Consider n points (or nodes) distributed uniformly and independently on the unit interval [0, 1]. Two nodes are said to be adjacent if their distance is less than some given threshold value. For the underlying random graph we derive zero-one laws for the property of graph connectivity and give the asymptotics of the transition widths for the associated phase transition. These results all flow from a single convergence statement for the probability of graph connectivity under a particular class of scalings. Given the importance of this result, we give two separate proofs; one approach relies on results concerning maximal spacings, while the other one exploits a Poisson convergence result for the number of breakpoint users.

Discrete Applied Mathematics, 2009

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph G n,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that G n,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of G n,k to our previous work on the connectivity of G n,k . Roughly speaking, we show that for s = o(log n), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, G n,k becomes s-connected very shortly after it becomes connected.

Random Structures and Algorithms, 2008

In this paper we study the one dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph.

IEEE Transactions on Information Theory, 2009

We consider a collection of n independent points which are distributed on the unit interval [0; 1] according to some probability distribution function F. Two nodes are said to be adjacent if their distance is less than some given threshold value. When F admits a nonvanishing density f, we show under a weak continuity assumption on f that the property of graph connectivity for the induced geometric random graph exhibits a strong zero-one law, and we identify the corresponding critical scaling. This is achieved by generalizing to nonuniform distributions a limit result obtained by Lévy for maximal spacings under the uniform distribution.

IEEE Communications Letters, 2007

We consider the geometric random graph where n points are distributed uniformly and independently on the unit interval [0, 1]. Using the method of first and second moments, we provide a simple proof of a very strong "zero-one" law for the property of graph connectivity under the asymptotic regime created by having n become large and the transmission range scaled appropriately with n.

Electronic Journal of Probability, 2013

We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit theorems. These conditions can always be expressed in terms of contraction operators or, equivalently, fourth cumulants. Our findings are specifically tailored to deal with the normal approximation of the geometric U -statistics introduced by Reitzner and Schulte (2011). In particular, we shall provide a new analytic characterization of geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussian fluctuations, and describe a new form of Poisson convergence for stationary random graphs with sparse connections. In a companion paper, the above analysis is extended to general sequences of U -statistics with rescaled kernels.

2007

We consider the geometric random graph where n points are distributed independently on the unit interval [0, 1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some transmission range. When F admits a continuous density f which is strictly positive on [0, 1], we show that the property of graph connectivity exhibits a strong critical threshold and we identify it. This is achieved by generalizing a limit result on maximal spacings due to Lévy for the uniform distribution.

Queueing Systems, 2012

We consider a collection of n independent points which are distributed on the unit interval [0, 1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some given threshold value. When F admits a density f which is strictly positive on [0, 1], we give conditions on f under which the property of graph connectivity for the induced geometric random graph obeys a very strong zero-one law when the transmission range is scaled appropriately with n large. The very strong critical threshold is identified. This is done by applying a version of the method of first and second moments. Keywords One-dimensional geometric random graphs • Connectivity • Non-uniform node distribution • Very strong zero-one laws This work was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U.S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.

The expected number of occurrences of a graph G as a subgraph of a random unit disc graph is expressed as a summation (related to a certain zeta function) over integer-valued circulations in a certain extension of G. In the special case where G is a path graph, this yields an interesting equality between three integral formulas.

2006

For a measure µ supported on a compact connected subset of a Euclidean space which satisfies a uniform d-dimensional decay of the volume of balls of the type αδ d ≤ µ(B(x, δ)) ≤ βδ d (1) we show that the maximal edge in the minimum spanning tree of n indepndent samples from µ is, with high probability ≈ log n n 1/d .